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A decreasing curve with an arrow at the end is plotted on a coordinate plane. The y-axis is labeled amount left parenthesis milligrams right parenthesis. The x-axis is labeled time left parenthesis hours right parenthesis.
Maria took 500 mg of medicine for her headache. The drug depletes in the blood stream at a rate of 20% per hour. The situation can be modeled by the exponential equation a(t)=500(0.8)t
, where a is the amount of medicine in the blood stream after t hours. Which one of the following statements is true about the graph of the function?
Option #1: As t increases, a increases slowly at first and then quickly.
Option #2: The y-intercept of the function is (0,500)
.
Option #3: The range includes all real numbers.
(1 point)
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The y-axis is labeled amount left parenthesis milligrams right parenthesis. The x-axis is labeled time left parenthesis hours right parenthesis.
Maria took 500 mg of medicine for her headache. The drug depletes in the blood stream at a rate of 20% per hour. The situation can be modeled by the exponential equation a(t)=500(0.8)t
, where a is the amount of medicine in the blood stream after t hours. Which one of the following statements is true about the graph of the function?
Option #1: As t increases, a increases slowly at first and then quickly.
Option #2: The y-intercept of the function is (0,500)
.
Option #3: The range includes all real numbers.
(1 point)
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To evaluate the truth of the statements given about the graph of the function \( a(t) = 500(0.8)^t \), let's analyze each option:
**Option #1:** "As t increases, a increases slowly at first and then quickly."
- This statement is not true. The function \( a(t) = 500(0.8)^t \) is a decreasing exponential function. As time \( t \) increases, the value of \( a(t) \) decreases, not increases. Therefore, this option is false.
**Option #2:** "The y-intercept of the function is (0,500)."
- To find the y-intercept, we evaluate \( a(0) \):
\[
a(0) = 500(0.8)^0 = 500(1) = 500
\]
Thus, the y-intercept is indeed \( (0, 500) \). Therefore, this option is true.
**Option #3:** "The range includes all real numbers."
- The range of \( a(t) = 500(0.8)^t \) is \( (0, 500] \). As \( t \) increases, \( a(t) \) approaches 0 but never reaches negative values. Hence, this option is false.
Based on this analysis, the only true statement about the graph of the function is **Option #2**: "The y-intercept of the function is (0,500)."
**Option #1:** "As t increases, a increases slowly at first and then quickly."
- This statement is not true. The function \( a(t) = 500(0.8)^t \) is a decreasing exponential function. As time \( t \) increases, the value of \( a(t) \) decreases, not increases. Therefore, this option is false.
**Option #2:** "The y-intercept of the function is (0,500)."
- To find the y-intercept, we evaluate \( a(0) \):
\[
a(0) = 500(0.8)^0 = 500(1) = 500
\]
Thus, the y-intercept is indeed \( (0, 500) \). Therefore, this option is true.
**Option #3:** "The range includes all real numbers."
- The range of \( a(t) = 500(0.8)^t \) is \( (0, 500] \). As \( t \) increases, \( a(t) \) approaches 0 but never reaches negative values. Hence, this option is false.
Based on this analysis, the only true statement about the graph of the function is **Option #2**: "The y-intercept of the function is (0,500)."
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